Optimal Control of a Continuous Stirred-Tank Reactor
Optimal Control of a Continuous Stirred-Tank Reactor
Aris and Amundson [1–3] originally analyzed the design of a continuous stirred-tank reactor (CSTR) with a first-order, irreversible exothermic reaction. The reactor is controlled by varying the flow of a cooling fluid through a coil inside the reactor. The state equations for the CSTR are given by
dT
dt
25T
T+2
u
1
dC
dt
25T
T+2
where and are the deviations from steady-state concentration and temperature, respectively, and is the control action.
C
T
u
1
This Demonstration finds the optimal control, which minimizes the performance index given by , where the normalized control action is and is a weighting parameter. Assume a final time =0.78 of the control [4, 5]. You can set the initial conditions (i.e. both and ) and choose to apply upper and lower bounds on .
J=[++R]dt
t
f
∫
0
2
T
2
C
2
u
u=-1
u
1
R=0.1
t
f
T(t=0)
C(t=0)
u
We plot the component of the state vector ( and shown in red and blue, respectively), the normalized control , and the costates. We also compute the value of the performance index .
T
C
u
J
For the case where is unbounded, if you select the initial conditions and , then , in close agreement with the result found in [4] and [5].
u
T(t=0)=0.05
C(t=0)=0
J=0.02660
The costates (t) and (t) are plotted in magenta and cyan. The costates verify ()=0 and ()=0 (yellow dot).
λ
1
λ
2
λ
1
t
f
λ
2
t
f